Arithmetic derivative perfect number

Question

Is there another perfect number $n$ besides 6 such that $n'<n$, where $n'$ is the arithmetic derivative?

Answer 1

$6$ is the only such even number.

The only even perfect numbers are $2^{p-1}(2^p-1)$ where $2^p-1$ is prime.

Suppose $p\ge3$, the derivative is then $(p-1)2^{p-2}(2^p-1)+2^{p-1}$ and this is greater than $2^{p-1}(2^p-1)$.